Over-Mahonian numbers: Basic properties and unimodality

TL;DR

This paper introduces over-Mahonian numbers counting overlined permutations with inversions, explores their properties through combinatorial models, and proves their unimodality and log-concavity.

Contribution

It defines over-Mahonian numbers, provides combinatorial interpretations, and proves their unimodality and log-concavity, extending classical permutation statistics.

Findings

Over-Mahonian numbers are unimodal and log-concave sequences.

Combinatorial interpretations include lattice paths, overpartitions, and tilings.

The paper establishes foundational properties of over-Mahonian numbers.

Abstract

In this paper, we introduce the concept of the over-Mahonian number, which counts the overlined permutations of length $n$ with $k$ inversions, allowing the first elements associated with the inversions to be independently overlined or not. We explore its properties and combinatorial interpretations through lattice paths, overpartitions, and tilings, and provide a combinatorial proof demonstrating that these numbers form a log-concave and unimodal sequence.